In this paper we propose a multiscale method for the acoustic wave equation in highly oscillatory media. We use a higher order extension of the localized orthogonal decomposition method combined with a higher order time stepping scheme and present rigorous a priori error estimates in the energy-induced norm. We find that in the very general setting without additional assumptions on the coefficient beyond boundedness arbitrary orders of convergence cannot be expected, but that increasing the polynomial degree may still considerably reduce the size of the error. Under additional regularity assumptions higher orders can be obtained as well. Numerical examples are presented that confirm the theoretical results.

中文翻译：

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波动方程的高阶多尺度方法


在本文中，我们提出了一种用于高振荡介质中声波方程的多尺度方法。我们使用局部正交分解方法的高阶扩展与高阶时间步进方案相结合，并在能量诱导范数中提出严格的先验误差估计。我们发现，在非常一般的设置中，如果没有对超出有界的系数进行额外的假设，则无法预期任意阶的收敛，但是增加多项式次数仍然可以大大减少误差的大小。在额外的规律性假设下，也可以获得更高的阶数。给出的数值例子证实了理论结果。